Circles

Let S be the set of all circles on the (cartesian) plane, with rules of addition and multiplication by scalars defined as follows.

If C1 (resp. C2) is a circle of center (x1,y1) (resp. (x2,y2) ) and radius , C1 + C2 will be the circle of center (x1+x2,y1+y2) and radius .

If C is a circle of center (x,y) and radius , and if a is a real number, then aC is a circle of center (ax,ay) and radius .

Is S with the addition and multiplication by scalar defined above is a vector space over the field of real numbers?

Space of maps

Let S be the set of maps

f: ---> ,

(i.e., from the set of to the set of ) with rules of addition and multiplication by scalar as follows:

If f1 and f2 are two maps in S, f1+f2 is a map f: : -> such that f(x)=f1(x)+f2(x) for all x belonging to .

If f is a map in S and if a is a real number, af is a map from to such that (af)(x)=a·f(x) for all x belonging to .

Is S with the structure defined above is a vector space over R ?

Absolute value

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:

For any (x,y) and (x,y) belonging to S, we define (x,y)+(x,y) = (x+x,y+y).

For any (x,y) belonging to S and any real number a, we define a(x,y) = (|a|x,|a|y).

Is S with the structure defined above is a vector space over R?

Affine line

Let L be a line in the cartesian plane, defined by an equation c1x+c2y=c3, and let =(x,y) be a fixed point on L.

We take S to be the set of points on L. On S, we define addition and multiplication by scalar as follows.

If =(x,y) and =(x,y) are two elements of S, we define + = .

If =(x,y) is an element of S and if is a real number, we define = .

Is S with the structure defined above is a vector space over R?

Alternated addition

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:

For any (x,y) and (x,y) belonging to S, (x,y)+(x,y) = (x+y,y+x).

For any (x,y) belonging to S and any real number a, a(x,y) = (ax,ay).

Is S with the structure defined above is a vector space over R?

Fields

The set of all , together with the usual addition and multiplication, is it a vector space over the field of ?

Matrices

Let
be the set of real
matrices. On
, we define the multiplicatin by scalar as follows. If
is a matrix in
, and if
is a real number, the multiplication of
by the scalar
is defined to be the matrix
, where
.

Is
together with the usual addition and the above multiplication by scalar a vector space over
?

Matrices II

The set of matrices of elements and of , together with the usual addition and multiplication, is it a vector space over the field of ?

Multiply/divide

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:

For any (x,y) and (x,y) belonging to S, we define (x,y)+(x,y) = (x+x,y+y).

For any (x,y) belonging to S and any real number a, we define a(x,y) = (x/a , y/a) if a is non-zero, and 0(x,y)=(0,0).

Is S with the structure defined above is a vector space over R?

Non-zero numbers

Let S be the set of real numbers. We define addition and multiplication by scalare on S as follows:

If x and y are two elements of S, the sum of x and y in S is defined to be xy.

If x is an element of S and if a is a real number, the multiplication of x by the scalare a is defined to be xa.

Is S with the structure defined above is a vector space over R?

Transaffine

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:

If (x,y) and (x,y) are two elements of S, their sum in S is defined to be the couple (x+x,y+y).

If (x,y) is an element of S, and if a is a real number, the multiplication of (x,y) by the scalar a in S is defined to be the couple (ax(),ay()).

Is S with the structure defined above is a vector space over R?

Transquare

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:

For any (x,y) and (x,y) belonging to S, (x,y)+(x,y) = (x+x,y+y).

For any (x,y) belonging to S and any real number a, a(x,y) = (ax,ay()2).

Is S with the structure defined above is a vector space over R?

Unit circle

Let S be the set of points on the circle x2+y2=1 in the cartesian plane. For any point (x,y) in S, there is a real number t such that x=cos(t), y=sin(t).

We define the addition and multiplication by scalare on S as follows:

If (cos(t1),sin(t1)) and (cos(t2),sin(t2)) are two points in S, their sum is defined to be (cos(t1+t2),sin(t1+t2)).

If p=(cos(t), sin(t)) is a point in S and if a is a real number, the multiplication of p by the scalar a is defined to be (cos(at), sin(at)).